12.10.2019

Total surface area and volume of a cone. The area of the lateral and full surface of the cone

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Lesson type: a lesson in studying new material using elements of a problem-developing teaching method.

Lesson Objectives:

cognitive:
- familiarization with a new mathematical concept;
- formation of new ZUN;
- the formation of practical skills for solving problems.
developing:
- development of independent thinking of students;
- skills development correct speech schoolchildren.
educational:
- development of teamwork skills.

Lesson equipment: magnetic board, computer, screen, multimedia projector, cone model, lesson presentation, handout.

Lesson objectives (for students):

get acquainted with a new geometric concept - a cone;
derive a formula for calculating the surface area of a cone;
learn to apply the acquired knowledge in solving practical problems.

During the classes

I stage. Organizational.

Submission of notebooks with home test work on the topic covered.

Students are invited to find out the topic of the upcoming lesson by solving the rebus (slide 1):

Picture 1.

Announcement to students of the topic and objectives of the lesson (slide 2).

II stage. Explanation of new material.

1) Teacher's lecture.

On the board is a table with the image of a cone. new material explained in the accompanying program material "Stereometry". A three-dimensional image of a cone appears on the screen. The teacher gives a definition of a cone, talks about its elements. (slide 3). It is said that a cone is a body formed by the rotation of a right triangle relative to the leg. (slides 4, 5). An image of the development of the lateral surface of the cone appears. (slide 6)

2) Practical work.

Update basic knowledge: repeat the formulas for calculating the area of a circle, the area of a sector, the circumference of a circle, the length of an arc of a circle. (slides 7-10)

The class is divided into groups. Each group receives a scan of the lateral surface of the cone cut out of paper (a circle sector with an assigned number). Students take the necessary measurements and calculate the area of the resulting sector. Instructions for doing work, questions - problem statements - appear on the screen (slides 11-14). The representative of each group writes the results of the calculations in a table prepared on the board. The participants of each group glue the model of the cone from the development they have. (slide 15)

3) Statement and solution of the problem.

How to calculate the lateral surface area of a cone if only the radius of the base and the length of the generatrix of the cone are known? (slide 16)

Each group makes the necessary measurements and tries to derive a formula for calculating the required area using the available data. When doing this work, students should notice that the circumference of the base of the cone is equal to the length of the arc of the sector - the development of the lateral surface of this cone. (slides 17-21) Using the necessary formulas, the desired formula is derived. Students' reasoning should look something like this:

The radius of the sector - sweep is equal to l, the degree measure of the arc is φ. The area of the sector is calculated by the formula: the length of the arc bounding this sector is equal to the Radius of the base of the cone R. The length of the circle lying at the base of the cone is C = 2πR. Note that Since the area of the lateral surface of the cone is equal to the area of the development of its lateral surface, then

So, the area of the lateral surface of the cone is calculated by the formula S BOD = πRl.

After calculating the lateral surface area of the cone model according to the formula derived independently, a representative of each group writes the result of the calculations in a table on the board in accordance with the model numbers. The calculation results in each row must be equal. On this basis, the teacher determines the correctness of the conclusions of each group. The result table should look like this:

model no.	I task	II task

	(125/3)π ~ 41.67π

	(425/9)π ~ 47.22π
	(539/9)π ~ 59.89π

Model parameters:

l=12 cm, φ=120°
l=10 cm, φ=150°
l=15 cm, φ=120°
l=10 cm, φ=170°
l=14 cm, φ=110°

The approximation of calculations is associated with measurement errors.

After checking the results, the output of the formulas for the areas of the lateral and full surfaces of the cone appears on the screen (slides 22-26) students keep notes in notebooks.

III stage. Consolidation of the studied material.

1) Students are offered tasks for oral solution on ready-made drawings.

Find the areas of the total surfaces of the cones shown in the figures (slides 27-32).

2) Question: Are the areas of the surfaces of cones formed by the rotation of one right triangle about different legs equal? Students make a hypothesis and test it. Hypothesis testing is carried out by solving problems and is written by the student on the blackboard.

Given:Δ ABC, ∠C=90°, AB=c, AC=b, BC=a;

BAA", ABV" - bodies of revolution.

Find: S PPC 1 , S PPC 2 .

Figure 5 (slide 33)

Solution:

1) R=BC = a; S PPC 1 = S BOD 1 + S main 1 = π a c + π a 2 \u003d π a (a + c).

2) R=AC = b; S PPC 2 = S BOD 2 + S main 2 = π b c + π b 2 \u003d π b (b + c).

If S PPC 1 = S PPC 2, then a 2 + ac \u003d b 2 + bc, a 2 - b 2 + ac - bc \u003d 0, (a-b) (a + b + c) \u003d 0. Because a, b, c positive numbers (the lengths of the sides of the triangle), the tore-equality is true only if a =b.

Conclusion: The areas of the surfaces of two cones are equal only if the legs of the triangle are equal. (slide 34)

3) Solution of the problem from the textbook: No. 565.

IV stage. Summing up the lesson.

Homework: p.55, 56; No. 548, No. 561. (slide 35)

Announcement of grades.

Conclusions during the lesson, repetition of the main information received in the lesson.

Literature (slide 36)

Geometry grades 10–11 - Atanasyan, V. F. Butuzov, S. B. Kadomtsev et al., M., Enlightenment, 2008.
"Mathematical puzzles and charades" - N.V. Udaltsov, library "First of September", series "MATHEMATICS", issue 35, M., Chistye Prudy, 2010.

The bodies of revolution studied at school are a cylinder, a cone and a ball.

If in a USE task in mathematics you need to calculate the volume of a cone or the area of a sphere, consider yourself lucky.

Apply formulas for volume and surface area of a cylinder, cone, and sphere. All of them are in our table. Learn by heart. This is where the knowledge of stereometry begins.

Sometimes it's good to draw a top view. Or, as in this problem, from below.

2. How many times greater is the volume of a cone circumscribed near a regular quadrangular pyramid than the volume of a cone inscribed in this pyramid?

Everything is simple - we draw a view from below. We see that the radius of the larger circle is several times larger than the radius of the smaller one. The heights of both cones are the same. Therefore, the volume of the larger cone will be twice as large.

Another important point. Remember that in the tasks of part B USE options in mathematics, the answer is written as an integer or finite decimal fraction. Therefore, you should not have any or in your answer in part B. Substituting the approximate value of the number is also not necessary! It must be reduced! It is for this that in some tasks the task is formulated, for example, as follows: "Find the area of the lateral surface of the cylinder divided by".

And where else are the formulas for the volume and surface area of bodies of revolution used? Of course, in problem C2 (16). We will also tell you about it.

We know what a cone is, let's try to find its surface area. Why is it necessary to solve such a problem? For example, you need to understand how much dough will go to make a waffle cone? Or how many bricks would it take to lay down the brick roof of a castle?

It is not easy to measure the lateral surface area of a cone. But imagine the same horn wrapped in cloth. To find the area of a piece of fabric, you need to cut it and lay it out on the table. We get a flat figure, we can find its area.

Rice. 1. Section of the cone along the generatrix

Let's do the same with the cone. Let's cut it side surface along any generatrix, for example, (see Fig. 1).

Now we “unwind” the side surface onto a plane. We get a sector. The center of this sector is the top of the cone, the radius of the sector is equal to the generatrix of the cone, and the length of its arc coincides with the circumference of the base of the cone. Such a sector is called a development of the lateral surface of the cone (see Fig. 2).

Rice. 2. Development of the side surface

Rice. 3. Angle measurement in radians

Let's try to find the area of the sector according to the available data. First, let's introduce a notation: let the angle at the top of the sector be in radians (see Fig. 3).

We will often encounter the angle at the top of the sweep in tasks. In the meantime, let's try to answer the question: can't this angle turn out to be more than 360 degrees? That is, will it not turn out that the sweep will superimpose itself? Of course not. Let's prove it mathematically. Let the sweep "overlap" itself. This means that the length of the sweep arc is greater than the circumference of the radius . But, as already mentioned, the length of the sweep arc is the circumference of the radius. And the radius of the base of the cone, of course, is less than the generatrix, for example, because the leg of a right triangle is less than the hypotenuse

Then let's remember two formulas from the course of planimetry: arc length. Sector area: .

In our case, the role is played by the generatrix , and the length of the arc is equal to the circumference of the base of the cone, that is. We have:

Finally we get:

Along with the lateral surface area, the total surface area can also be found. To do this, add the base area to the lateral surface area. But the base is a circle of radius , whose area, according to the formula, is .

Finally we have: , where is the radius of the base of the cylinder, is the generatrix.

Let's solve a couple of problems on the given formulas.

Rice. 4. Desired angle

Example 1. The development of the lateral surface of the cone is a sector with an angle at the apex. Find this angle if the height of the cone is 4 cm and the radius of the base is 3 cm (see Fig. 4).

Rice. 5. Right triangle forming a cone

By the first action, according to the Pythagorean theorem, we find the generatrix: 5 cm (see Fig. 5). Further, we know that .

Example 2. The area of the axial section of the cone is , the height is . Find the total surface area (see Fig. 6).